Bringing You The Future Of Science and Technology

WEIN'S DISPLACEMENT LAW

For this tutorial, we will learn more about Wein's Displacement Law and how it can be used to calculate the dominant wavelength exiting a blackbody radiator. We will use Python, Numpy and Matplotlib to plot the blackbody radiation curves for various celestial objects.

Categories

Art and Web Design

Latest Articles

Wein's Displacement Law

Wien's displacement law can be used to describe the relationship between the true temperature of a blackbody (T) in degrees Kelvin and its peak spectral exitance, or dominant wavelength (λmax). The dominant wavelength of an object is important to thermal remote sensing and can be used to classify stars based on their temperature.

λmax = k / T
k = 2892 μm
T = temperature of object (in Kelvin)

where k is a constant equaling 2898 μm K. We can determine the dominant wavelength of any object by substituting its temperature into the above equation.

There is a shift from longer to shorter wavelengths as the temperature of the blackbody increases. A blackbody is a hypothetical object, first described by Max Planck in 1900, that absorbs all incident electromagnetic radiation while still maintaining thermal equilibrium. This suggests that light is not reflected from or passes through the object, but radiation is emitted from the object, this is called blackbody radiation.

We can use this knowledge to determine the dominant wavelength of a particular star, based on its temperature, and accurately determine it's color based on it's peak spectral exitance. This is because stars approximate blackbody radiators, whose visible colors depend on the temperature of the radiator. When the temperature of a blackbody radiator increases, the radiant exitant energy increases and the peak of the radiation curve therefore moves to shorter wavelengths.

Our sun, for example, is a main sequence (G2V type) yellow dwarf star about 6000 K at it's surface. The dominant wavelength for a 6000 K star is 0.48 μm (480 nm). We can calculate the Sun's dominant wavelength using Python.

Next we will plot the blackbody radiation curves for various stars, ranging in temperature from 4000 K to 7000 K using Python, Numpy and Matplotlib.

Blackbody Radiation Curve

Stars are blackbody radiators whose visible color depends of the temperature of the radiator. From the dominant wavelength, we can approximate a star's temperature using Wein's displacement law.

Before we begin calculating the blackbody radiation curves, let us gather some information. The below table shows each band of the electromagnetic spectrum, with their related wavelengths and frequencies. This information will be useful when we calculate the dominant wavelength of various stars, as it tells us the approximate color of each star based on their temperature.

Band

Wavelength

Frequency

Gamma Rays

< 10 pm

> 30 EHz

X Rays

0.01 - 10 nm

30 EHz - 30 PHz

UV

10 - 400 nm

30 PHz - 790 THz

Violet

380 - 450 nm

668 - 789 THz

Blue

450 - 495 nm

606 - 668 THz

Green

495 - 570 nm

526 - 606 THz

Yellow

570 - 590 nm

508 - 526 THz

Orange

590 - 620 nm

484 - 508 THz

Red

620 - 750 nm

400 - 484 THz

Near IR

0.7 μm - 1.4 μm

214 - 400 THz

Short IR

1.4 - 3 μm

30 PHz - 790 THz

Mid IR

3 - 8 μm

30 PHz - 790 THz

Long IR

8 - 15 μm

30 PHz - 790 THz

Far IR

15 - 1000 μm

30 PHz - 790 THz

Microwaves

1 mm - 1 m

300 GHz - 300 MHz

Radoiwaves

1 m - 100000 km

300 MHz - 3 Hz

Next, make sure that you have properlly installed the Numpy and Matplotlib libraries. You can do this for both using the pip install command.

$ pip install numpy
$ pip install matplotlib

Now we can build our application. You can download the following code from my Github repository.

Executing this script creates in the following diagram, which shows the dominant wavelength of various stars ranging in temperature from 4000 K to 7000 K.

These blackbody radiation curves, also known as Planck Curves, describe the continuous spectra for stellar interiors. The outward appearance of stars depends more on the underlying continuous spectrum coming from the inner parts of a star than the absorption at its surface. The following table lists corresponding values of color, spectral type, temperature and peak wavelength (λmax). Note that we are talking about the surface temperature of a star.

Classification

Temperature

λmax

Color

O0

40,000 K

72.5 nm

Blue

B0

20,000 K

145 nm

Light Blue

A0

10,000 K

290 nm

White

F0

7,500 K

387 nm

Yellow-White

G0

5,500 K

527 nm

Yellow

K0

4,000 K

725 nm

Orange

M0

3,000 K

966 nm

Red

Remote Sensing Applications

The relationship between the temperature of a blackbody and it's peak spectral exitance is known as Wein's displacement law, and is useful for determining the temperature of radiant objects such as stars. This can also be used to determine the temperature of any radiant object whose temperature exceeds that of its surroundings. The dominant wavelength provides valuable information regarding parts of the thermal infrared spectrum in which we want to sense the object.

From a remote sensing perspective, we would consider the temperature ranges of the phenomenon which we hope to detect. For example, if we are observing 800 K forest fires with a dominant wavelength of approximately 3.62 μm, then we would choose a remote sensing system that uses a 3 - 5 μm thermal infrared detector. Likewise, if we wanted to observe the soil, water or other surface object, which is approximately 300 K with a dominant wavelength of 9.67 μm, then we would pick a thermal infrared detector that operates in the 8 - 14 μm region.